Is there a single verb or noun synonymous with "pursuit of pleasure" either in English, Latin or (Ancient) Greek? Is there such a single verb or noun for "avoidance and prevention of pain"?
I'd go back to αἵρεσις (hairesis) and φεύγω (pheugo). αἵρεσις can mean the taking of a town in a battle; choice or election of magistrates. Liddell & Scott write that its opposite is some senses is κλῆρος (kleros) which is the casting of lots. So it's the difference between making an informed choice (αἵρεσις) or making a decision by flipping a coin (κλῆρος). φεύγω can be thought of as people fleeing that town that's being taken in a battle; they're escaping from their fate; they have agency in fleeing the situation. The opposite of that word is διώκω (dioko) which is defined as pursue, chase, in war or hunting; pursue an object, seek after; or even drive or chase away.
Both αἵρεσις (hairesis) and φεύγω (pheugo), to me, convey agency in whatever direction one heads. It is not a passive activity, but one taken with vigor and purpose.